Is this statement true or false if false a counterexample is needed if true then an explanation The product AB of two square matrices can only have an inverse if A and B both have inverses.
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A matrix specifies a linear transformation. If a matrix is (or is not) invertible, what does that tell you about the rank of the associated linear transformation? Multiplying the matrices corresponds to composing the transformations.
Another way of thinking about it- a matrix is invertible if and only if its determinant is non-zero. And det(AB)= det(A)det(B).
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